This is the exact idea I came to suggest . But instead of a notification, I prefer a popup bubble that is draggable. Tapping on this bubble will bring in the Better Open With window with the list of other apps that can open the file.
The bubble should also have a timeout after which it will disappear.

That's how I understand it and just want I want. Of course, this would be an option that the user can choose to have it operate this way OR the default way (how it works now), preferably for each file type. And it could be a notification (have to pull down the notifications to access it) or, as I would prefer it, a toast notification. And the delay before it goes away as well as the location (top vs bottom) could be user-defined. If that's too much to have all those options defined by the user, then I suggest just using your own judgement as I'm sure however you decide to do it will be fine and it will just be nice to have this option.

If I understand correctly, you want BOW to do this:
- You open a file (e.g. a PDF)
- BOW launches the default app without seeing the list
- You see a notification that would launch BOW's dialogue screen to open the file with a different app